Models Of Conformal Field Theory Research Articles

Double-soft behavior of the dilaton of spontaneously broken conformal invariance

The Ward identities involving the currents associated to the spontaneously broken scale and special conformal transformations are derived and used to determine, through linear order in the two soft-dilaton momenta, the double-soft behavior of scattering amplitudes involving two soft dilatons and any number of other particles. It turns out that the double-soft behavior is equivalent to performing two single-soft limits one after the other. We confirm the new double-soft theorem perturbatively at tree-level in a D-dimensional conformal field theory model, as well as nonperturbatively by using the “gravity dual” of mathcal{N}=4 super Yang-Mills on the Coulomb branch; i.e. the Dirac-Born-Infeld action on AdS5 × S5.

Entanglement in Nonunitary Quantum Critical Spin Chains.

Entanglement entropy has proven invaluable to our understanding of quantum criticality. It is natural to try to extend the concept to "nonunitary quantum mechanics," which has seen growing interest from areas as diverse as open quantum systems, noninteracting electronic disordered systems, or nonunitary conformal field theory (CFT). We propose and investigate such an extension here, by focusing on the case of one-dimensional quantum group symmetric or supergroup symmetric spin chains. We show that the consideration of left and right eigenstates combined with appropriate definitions of the trace leads to a natural definition of Rényi entropies in a large variety of models. We interpret this definition geometrically in terms of related loop models and calculate the corresponding scaling in the conformal case. This allows us to distinguish the role of the central charge and effective central charge in rational minimal models of CFT, and to define an effective central charge in other, less well-understood cases. The example of the sl(2|1) alternating spin chain for percolation is discussed in detail.

Flat structures on the deformations of Gepner chiral rings

We propose a simple method for the computation of the flat coordinates and Saito primitive forms on Frobenius manifolds of the deformations of Jacobi rings associated with isolated singularities. The method is based on using a conjecture about integral representations for the flat coordinates and on the Saito cohomology theory. This reduces the computation to a simple linear problem. We consider the case of the deformed Gepner chiral rings. The knowledge of the flat structures of Frobenius manifolds can be used for exact solution of the models of the topological conformal field theories corresponding to these chiral rings.

OndS4extremal surfaces and entanglement entropy in some ghost CFTs

In arXiv:1501.03019 [hep-th], the areas of certain complex extremal surfaces in de Sitter space were found to have resemblance with entanglement entropy in appropriate dual Euclidean non-unitary CFTs, with the area being real and negative in $dS_4$. In this paper, we study some toy models of 2-dim ghost conformal field theories with negative central charge with a view to exploring this further from the CFT point of view. In particular we consider $bc$-ghost systems with central charge $c=-2$ and study the replica formulation for entanglement entropy for a single interval, and associated issues arising in this case, notably pertaining to (i) the $SL(2)$ vacuum coinciding with the ghost ground state, and (ii) the background charge inherent in these systems which leads to particular forms for the norms of states (involving zero modes). This eventually gives rise to negative entanglement entropy. We also discuss a (logarithmic) CFT of anti-commuting scalars, with similarities in some features. Finally we discuss a simple toy model of two "ghost-spins" which mimics some of these features.

Scaling and superscaling solutions from the functional renormalization group

We study the renormalization group flow of $\mathbb{Z}_2$-invariant supersymmetric and non-supersymmetric scalar models in the local potential approximation using functional renormalization group methods. We focus our attention to the fixed points of the renormalization group flow of these models, which emerge as scaling solutions. In two dimensions these solutions are interpreted as the minimal (supersymmetric) models of conformal field theory, while in three dimension they are manifestations of the Wilson-Fisher universality class and its supersymmetric counterpart. We also study the analytically continued flow in fractal dimensions between 2 and 4 and determine the critical dimensions for which irrelevant operators become relevant and change the universality class of the scaling solution. We also include novel analytic and numerical investigations of the properties that determine the occurrence of the scaling solutions within the method. For each solution we offer new techniques to compute the spectrum of the deformations and obtain the corresponding critical exponents.

The fusion rules for the Temperley–Lieb algebra and its dilute generalization

The Temperley–Lieb (TL) family of algebras is well known for its role in building integrable lattice models. Even though a proof is still missing, it is agreed that these models should go to conformal field theories in the thermodynamic limit and that the limiting vector space should carry a representation of the Virasoro algebra. The fusion rules are a notable feature of the Virasoro algebra. One would hope that there is an analogous construction for the TL family. Such a construction was proposed by Read and Saleur (2007 Nucl. Phys. B 777 316) and partially computed by Gainutdinov and Vasseur (2013 Nucl. Phys. B 868 223–70) using the bimodule structure over the TL algebras and the quantum group Uq (sl2).We use their definition for the dilute Temperley–Lieb (dTL) family, a generalization of the original TL family. We develop a new way of computing fusion by using induction and show its power by obtaining fusion rules for both dTL and TL. We recover those computed by Gainutdivov and Vasseur and new ones that were beyond their scope. In particular, we identify a set of irreducible TL- or dTL-representations whose behavior under fusion is that of some irreducibles of the minimal models of conformal field theory.

Form factor relocalisation and interpolating renormalisation group flows from the staircase model

We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering.

Invariant functionals and Zamolodchikovs’ integral

A complex Selberg integral appearing in the Liouville model of conformal field theory is calculated by using the Bernstein-Reznikov method.

Kazama-Suzuki models of N = 2 superconformal field theory and Manin triples

Kazama-Suzuki coset models is an interesting class of N=2 supersymmetric models of conformal field theory which are used to build realistic models of superstring in 4 dimensions. We formulate Kazama-Suzuki construction of N=2 superconformal coset models using more general language of Manin triples and represent the corresponding N=2 Virasoro superalgebra currents in explicit form. A correspondence between the Kazama-Suzuki models and Poisson homogeneous spaces is established also.

A Solution Space for a System of Null-State Partial Differential Equations: Part 4

This article is the last of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Loewner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the first two articles, we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building on these results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $\mathcal{S}_N$. First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $8/\kappa$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of $\mathcal{S}_N$, which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE$_\kappa$ process join in a particular connectivity. Finally, we propose a reason for why the "exceptional speeds" (certain $\kappa$ values that appeared in the analysis of the Coulomb gas solutions in the third article) and the minimal models of CFT are connected.

Conformal blocks, Berenstein–Zelevinsky triangles, and group-based models

Work of Buczynska, Wiśniewski, Sturmfels and Xu, and the second author has linked the group-based phylogenetic statistical model associated with the group $$\mathbb {Z}/2\mathbb {Z}$$ Z / 2 Z with the Wess---Zumino---Witten (WZW) model of conformal field theory associated to $$\mathrm {SL}_2(\mathbb {C})$$ SL 2 ( C ) . In this article we explain how this connection can be generalized to establish a relationship between the phylogenetic statistical model for the cyclic group $$\mathbb {Z}/m\mathbb {Z}$$ Z / m Z and the WZW model for the special linear group $$\mathrm {SL}_m(\mathbb {C}).$$ SL m ( C ) . We use this relationship to also show how a combinatorial device from representation theory, the Berenstein---Zelevinsky triangle, corresponds to elements in the affine semigroup algebra of the $$\mathbb {Z}/3\mathbb {Z}$$ Z / 3 Z phylogenetic statistical model.

Global Gauge Anomalies in Coset Models of Conformal Field Theory

We study the occurrence of global gauge anomalies in the coset models of two-dimensional conformal field theory that are based on gauged WZW models. A complete classification of the non-anomalous theories for a wide family of gauged rigid adjoint or twisted-adjoint symmetries of WZW models is achieved with the help of Dynkin's classification of Lie subalgebras of simple Lie algebras.

The black hole interior and a curious sum rule

We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics.

Elliptic Hypergeometry of Supersymmetric Dualities II. Orthogonal Groups, Knots, and Vortices

We consider Seiberg electric-magnetic dualities for 4d $\mathcal{N}=1$ SYM theories with SO(N) gauge group. For all such known theories we construct superconformal indices (SCIs) in terms of elliptic hypergeometric integrals. Equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals. In particular, we describe a number of new elliptic beta integrals associated with the s-confining theories with the spinor matter fields. Reductions of some dualities from SP(2N) to SO(2N) or SO(2N+1) gauge groups are described. Interrelation of SCIs and the Witten anomaly is briefly discussed. Possible applications of the elliptic hypergeometric integrals to a two-parameter deformation of 2d conformal field theory and related matrix models are indicated. Connections of the reduced SCIs with the state integrals of the knot theory, generalized AGT duality for (3+3)d theories, and a 2d vortex partition function are described.

Logarithmic CFT models from Nichols algebras: I

We construct chiral algebras that centralize rank-2 Nichols algebras with at least one fermionic generator. This gives ‘logarithmic’ W-algebra extensions of a fractional-level algebra. We discuss crucial aspects of the emerging general relation between Nichols algebras and logarithmic conformal field theory (CFT) models: (i) the extra input, beyond the Nichols algebra proper, needed to uniquely specify a conformal model; (ii) a relation between the CFT counterparts of Nichols algebras connected by Weyl groupoid maps; and (iii) the common double bosonization U(X) of such Nichols algebras. For an extended chiral algebra, candidates for its simple modules that are counterparts of the U(X) simple modules are proposed, as a first step toward a functorial relation between U(X) and W-algebra representation categories.

The Algebra of $$ {{\mathbf{SL}}_3}\left( \mathbb{C} \right) $$ Conformal Blocks

We construct and study a family of toric degenerations of the Cox ring of the moduli of quasi-parabolic principal SL3( $$ \mathbb{C} $$ ) bundles on a smooth, marked curve (C, $$ \vec{p} $$ ): Elements of this algebra have a well known interpretation as conformal blocks, from the Wess-Zumino-Witten model of conformal field theory. For the genus 0; 1 cases we find the level of conformal blocks necessary to generate the algebra. In the genus 0 case we also find bounds on the degrees of relations required to present the algebra. As a consequence we obtain a toric degeneration for the projective coordinate ring of an effective divisor on the moduli $$ {{\mathcal{M}}_{{C,\vec{p}}}}\left( {\mathrm{S}{{\mathrm{L}}_3}\left( \mathbb{C} \right)} \right) $$ of quasi-parabolic principal SL3( $$ \mathbb{C} $$ ) bundles on (C, $$ \vec{p} $$ ). Along the way we recover positive polyhedral rules for counting conformal blocks.

PT -symmetric deformations of integrable models

We review recent results on new physical models constructed as -symmetrical deformations or extensions of different types of integrable models. We present non-Hermitian versions of quantum spin chains, multi-particle systems of Calogero–Moser–Sutherland type and nonlinear integrable field equations of Korteweg–de Vries type. The quantum spin chain discussed is related to the first example in the series of the non-unitary models of minimal conformal field theories. For the Calogero–Moser–Sutherland models, we provide three alternative deformations: a complex extension for models related to all types of Coxeter/Weyl groups; models describing the evolution of poles in constrained real-valued field equations of nonlinear integrable systems; and genuine deformations based on antilinearly invariant deformed root systems. Deformations of complex nonlinear integrable field equations of Korteweg–de Vries type are studied with regard to different kinds of -symmetrical scenarios. A reduction to simple complex quantum mechanical models currently under discussion is presented.

Fusion in the entwined category of Yetter-Drinfeld modules of a rank-1 Nichols algebra

We rederive a popular nonsemisimple fusion algebra in the braided context, from a Nichols algebra. Together with the decomposition that we find for the product of simple Yetter-Drinfeld modules, this strongly suggests that the relevant Nichols algebra furnishes an equivalence with the triplet W-algebra in the (p,1) logarithmic models of conformal field theory. For this, the category of Yetter-Drinfeld modules is to be regarded as an \textit{entwined} category (the one with monodromy, but not with braiding).

THE NICHOLS ALGEBRA OF SCREENINGS

Two related constructions are associated with screening operators in models of two-dimensional conformal field theory. One is a local system constructed in terms of the braided vector space X spanned by the screening species in a given CFT model and the space of vertex operators Y and the other is the Nichols algebra 𝔅(X) and the category of its Yetter–Drinfeld modules, which we propose as an algebraic counterpart, in a "braided" version of the Kazhdan–Lusztig duality, of the representation category of vertex-operator algebras realized in logarithmic CFT models.

Algebraic properties of CFT coset construction and Schramm-Loewner evolution

Schramm-Loewner evolution appears as the scaling limit of interfaces in lattice models at critical point. Critical behavior of these models can be described by minimal models of conformal field theory. Certain CFT correlation functions are martingales with respect to SLE. We generalize Schramm-Loewner evolution with additional Brownian motion on Lie group G to the case of factor space G/A. We then study connection between SLE description of critical behavior with coset models of conformal field theory. In order to be consistent such construction should give minimal models for certain choice of groups.